factor x^4-8x^2+7 — step-by-step solution

1) Let's factorize in

ℝ

x^{4} - 8 x^{2} + 7

⇥1.1) Let's set:

X = x^{2}

⇥You can rewrite the expression as:

X^{2} - 8 X + 7

⇥Given the quadratic form

a X^{2} + b X + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -8

and

c = 7

⇥

Δ = {(-8)}^{2} - 4 \cdot 1 \cdot 7

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-8)}^{2}

(-8) \cdot (-8)

64

⇥

Δ = 64 - 4 \cdot 7

⇥ⓘ Multiplications are done before additions and subtractions:

4 \cdot 7

28

⇥

Δ = 64 - 28

⇥ⓘ Start from

64

and move back

28

on the number line:

64 - 28

36

⇥

Δ = 36

⇥As

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

X_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-8) - \sqrt{36}}{2 \cdot 1}

⇥

X_{1} = \frac{8 - \sqrt{36}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{36}

6

;

6^{2}

36

⇥

X_{1} = \frac{8 - 6}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

8 - 6

2

⇥

X_{1} = \frac{2}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

⇥

X_{1} = 1

⇥The quadratic formula gives

X_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-8) + \sqrt{36}}{2 \cdot 1}

⇥

X_{2} = \frac{8 + \sqrt{36}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{36}

6

;

6^{2}

36

⇥

X_{2} = \frac{8 + 6}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

8 + 6

14

⇥

X_{2} = \frac{14}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{14}{2}

7

⇥

X_{2} = 7

⇥Factors identified:

(X - 1) \cdot (X - 7)

⇥1.2) Let's factorize in

ℝ

(x^{2} - 1) \cdot (x^{2} - 7)

⇥⇥1.2.1) Let's factor in

ℝ

(x^{2} - 1)

⇥⇥For

x^{2} - 1

, we recognize a difference of two squares:

x^{2} - 1^{2}

. Apply

a^{2} - b^{2} = (a - b) \cdot (a + b)

, with

a = x

and

b = 1

▷The factor

(x^{2} - 1)

factors as:

(x + 1) \cdot (x - 1)

⇥⇥1.2.2) Let's factor in

ℝ

(x^{2} - 7)